L(s) = 1 | + 3-s + 4·5-s − 4·7-s − 2·9-s + 5·11-s + 4·13-s + 4·15-s − 2·17-s − 4·21-s + 11·25-s − 5·27-s − 4·29-s − 8·31-s + 5·33-s − 16·35-s − 8·37-s + 4·39-s + 3·41-s + 8·43-s − 8·45-s − 12·47-s + 9·49-s − 2·51-s − 12·53-s + 20·55-s + 3·59-s − 4·61-s + ⋯ |
L(s) = 1 | + 0.577·3-s + 1.78·5-s − 1.51·7-s − 2/3·9-s + 1.50·11-s + 1.10·13-s + 1.03·15-s − 0.485·17-s − 0.872·21-s + 11/5·25-s − 0.962·27-s − 0.742·29-s − 1.43·31-s + 0.870·33-s − 2.70·35-s − 1.31·37-s + 0.640·39-s + 0.468·41-s + 1.21·43-s − 1.19·45-s − 1.75·47-s + 9/7·49-s − 0.280·51-s − 1.64·53-s + 2.69·55-s + 0.390·59-s − 0.512·61-s + ⋯ |
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s) \, L(s)\cr =\mathstrut & -\, \Lambda(2-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 92416 ^{s/2} \, \Gamma_{\C}(s+1/2) \, L(s)\cr =\mathstrut & -\, \Lambda(1-s) \end{aligned}\]
Particular Values
\(L(1)\) |
\(=\) |
\(0\) |
\(L(\frac12)\) |
\(=\) |
\(0\) |
\(L(\frac{3}{2})\) |
|
not available |
\(L(1)\) |
|
not available |
\(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
| $p$ | $F_p(T)$ |
---|
bad | 2 | \( 1 \) |
| 19 | \( 1 \) |
good | 3 | \( 1 - T + p T^{2} \) |
| 5 | \( 1 - 4 T + p T^{2} \) |
| 7 | \( 1 + 4 T + p T^{2} \) |
| 11 | \( 1 - 5 T + p T^{2} \) |
| 13 | \( 1 - 4 T + p T^{2} \) |
| 17 | \( 1 + 2 T + p T^{2} \) |
| 23 | \( 1 + p T^{2} \) |
| 29 | \( 1 + 4 T + p T^{2} \) |
| 31 | \( 1 + 8 T + p T^{2} \) |
| 37 | \( 1 + 8 T + p T^{2} \) |
| 41 | \( 1 - 3 T + p T^{2} \) |
| 43 | \( 1 - 8 T + p T^{2} \) |
| 47 | \( 1 + 12 T + p T^{2} \) |
| 53 | \( 1 + 12 T + p T^{2} \) |
| 59 | \( 1 - 3 T + p T^{2} \) |
| 61 | \( 1 + 4 T + p T^{2} \) |
| 67 | \( 1 - T + p T^{2} \) |
| 71 | \( 1 + 4 T + p T^{2} \) |
| 73 | \( 1 - 11 T + p T^{2} \) |
| 79 | \( 1 - 8 T + p T^{2} \) |
| 83 | \( 1 - 17 T + p T^{2} \) |
| 89 | \( 1 - 6 T + p T^{2} \) |
| 97 | \( 1 + 9 T + p T^{2} \) |
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\(L(s) = \displaystyle\prod_p \ \prod_{j=1}^{2} (1 - \alpha_{j,p}\, p^{-s})^{-1}\)
Imaginary part of the first few zeros on the critical line
−14.07428445911328, −13.55955189967688, −13.28739639720638, −12.70367944615711, −12.42440413311460, −11.53288893797100, −11.01123383655917, −10.63063281955284, −9.889082411369352, −9.399801927416897, −9.140494062916277, −9.000432650227537, −8.269893509371685, −7.432159254700772, −6.666100404314815, −6.412494509610688, −6.067283067249966, −5.561005575984678, −4.904019879958801, −3.839820940284052, −3.521417853422989, −3.060548707086055, −2.200399627475462, −1.815319042974392, −1.085903138853665, 0,
1.085903138853665, 1.815319042974392, 2.200399627475462, 3.060548707086055, 3.521417853422989, 3.839820940284052, 4.904019879958801, 5.561005575984678, 6.067283067249966, 6.412494509610688, 6.666100404314815, 7.432159254700772, 8.269893509371685, 9.000432650227537, 9.140494062916277, 9.399801927416897, 9.889082411369352, 10.63063281955284, 11.01123383655917, 11.53288893797100, 12.42440413311460, 12.70367944615711, 13.28739639720638, 13.55955189967688, 14.07428445911328